Frequency Dependence of Quantum Localization in a Periodically Driven System Manabu Machida, Keiji Sai to, and Seiji Miyashita Department of Applied Physics, The University of Tokyo
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Frequency Dependence of Quantum Localization
in a Periodically Driven System
Manabu Machida, Keiji Saito, and Seiji Miyashita
Department of Applied Physics, The University of Tokyo
Matrices of Gaussian Orthogonal Ensemble (GOE) are real symmetric, and each element of them is a Gaussian distributed random number. 2
,2 1,0 jiijij HH
GOE Random Matrix
E.P. Wigner introduced random matrices to Physics. Wigner, F.J. Dyson, and many other physicists developed random matrix theory.
VtHtH )()( 0
0H V
)sin()( tAt
and are independently created GOE random matrices.
is fixed at 0.5.
Hamiltonian
A varies.
Typical Hamiltonian for complexly interacting systems
under an external field.
/2
0
)(i
expT dttHF
iexpF
Floquet Theory
0 nn F
Energy after nth period:
We define,
Energy fluctuates around satE
0
0
HnTH
nT
Comparing
/
Saturated !
Solid line
Esat is normalized so that the ground state energy is 0 and the energy at the center of the spectrum is 1.
nn H satEwith
0.02
0.1
0.2
0.41.0
as a function of satE /
How to understand the localization?
(i) Independent Landau-Zener Transitions
Wilkinson considered the energy change of a random matrix system when the parameter is swept.
M. Wilkinson, J.Phys.A 21 (1988) 4021
M. Wilkinson, Phys.Rev.A 41 (1990) 4645
We assume transitions of states occur at avoided crossings by the Landau-Zener formula, and each transition takesplace independently.
How to understand the localization?
Transition probability
Probability of finding the state on the lth level
Diffusion equation:
X
The integral on the exponential diverges.
Therefore,
1sat E
Quantum interference effect is essential!
for any
How to understand the localization?
The global transition cannot be understood only by the Landau-Zener transition.
The random matrix system The Anderson localization
In each time interval T, the system evolves by the Floquet operator F.
The Hamiltonian which brings about the Anderson localization evolves in the interval T,
THU A
iexp
F
AH
How to understand the localization?
(ii) Analogy to the Anderson Localization
ApQp
'',
A iitiivHiii
i
: random potential distributed uniformly in the width W
:Hamiltonian for the Anderson localization
m
m
W
tpmp
AA
iv
mpmp QA
N
mm
m EpE1
)1(Qsat
How to understand the localization?
mPmP QQ
Let us introduce in order to study -dependence of the quantum localization.
F. Haake, M. Kus, and R. Scharf, Z.Phys.B 65 (1987) 381
K. Zyczkowski, J.Phys.A 23 (1990) 4427
min
1
2
0
2
0
2
0minmin 1;:min1 N
rNN
rN
We count the number of relevant Floquet states in the initial state.
minN
99.0r
One important aspect of the quantum localization
/2
0 e
-dependence of Nmin
ba
NN1/
11
*min
*min
Phenomenologically,
Parameters in the phenomenological function of minN
/1Q eP
m/QQ emPmP
minN
minN (numerical)
/2
0 e
/Q e hP
h
h : unknown amplitude
This fact suggests the local transition probability originates in the Landau-Zener transition.
The quantum localization occurs in this random matrix due to the quantum interference effect. On the other hand, the Landau-Zener mechanism still works in the local transitions.
To be appeared in J.Phys.Soc.Jpn. 71(2002)
Conclusion
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